We had "Miss Suzie", "Heigh ho, heigh ho, it's off to school we go", and "Great big globs of greasy grimey gopher guts" in southeastern Wisconsin in the 90's. Here's a few more:

(to sing on a school bus)
Hail to the bus driver, bus driver, bus driver
Hail to the bus driver, bus driver man!
He drinks and he cusses and stinks up the buses.
Hail to the bus driver, bus driver man!

(a few to figure out who's it, by tapping people's toes)
Train train number 9 going down Chicago line
If the train should jump the track, do you want your money back?
Y-E-S and that spells yes!
(or)
N-O and that spells no!

Mickey Mouse built a house, how many bricks did he use?
(the person who's toe you land on picks a number, then the person reciting the rhyme counts to that number, and the final person landed on is it)

Bubblegum, bubblegum in a dish, how many pieces do you wish?
(works the same as Mickey Mouse above)

Eenie, meenie, miny, moe
Catch a tiger by the toe
If he hollars let him go
Eenie, meenie, miny, moe
My mother says that you are not it
Not because you're dirty, not because you're clean
But just because you kissed a boy behind a ma-ga-zine!
(there was another alternate line about a "dirty dirty dishrag on your big toe", too)

(to be sung on busses)
Give me a long M! EMMMMMM!
Give me a short M! EM!
Don't give me no pop, no pop, don't give me no tea, no tea, just give me that milk, moo moo moo moo, Wisconsin milk, moo moo moo moo.
(On the moo moo moo moo part, one person would lock their fingers together and turn their hands palms outward with their thumbs hanging down, and the other person would "milk" the thumb-utters. They'd switch for the second moo moo moo moo)
(This repeats for all the letters of the word 'milk', then...)
Give me a long milk! CHOCOLATE!
Give me a short milk! SKIM!
Don't give me no pop, no pop, don't give me no tea, no tea, just give me that milk, moo moo moo moo, Wisconsin milk, moo moo moo moo.

(some Christmas themed ones)
Jingle Bells, Batman smells, Robin laid an egg
The batmobile lost its wheel and Joker got away (or took ballet)
Batman's in the kitchen, Robin's in the hall, Joker's in the bathroom, peeing on the wall...(there was more, but I don't remember the rest)

Joy to the world, the teacher's dead.
We barbacued her head!
What happened to the body? We flushed it down the potty!
And round and round it goes, and round and round it goes,
forever and ever around it goes!

(A Barney one)
I hate you, you hate me,
we chased Barney up a tree.
With a 44 caliper shot him in the head,
aren't you glad that Barney's dead.

(This one has a little tune to it, although I don't know what the tune's name is.)
There's a place in France where the naked ladies dance.
There's a hole in the wall where the men can see it all.
But they don't care because they're eating underwear.

(To pop the head off a dandelion, you recite this)
Mama had a baby and her head popped off!

(an old classic)
Comet, it makes your teeth turn green.
Comet, it tastes like gasoline (alternately: vaseline)
Comet, it makes you vomit,
So buy your comet, and vomit, today!

(Songs that never end)
This is the sone that never ends,
It goes on and on my friends (alternately: it was made to annoy my friends)
Some people started singing it no knowing what it was, and they'll continue singing it forever just because (repeats from beginning)

I know the song that gets on everybody's nerves
I know the song that gets on everybody's nerves
I know the song the gets on everybody's nerves
And this is how it goes (repeat from the beginning)